\newproblem{lay:4_1_19}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 4.1.19}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	If a mass $m$ is placed at the end of a spring, and if the mass is pulled downward and released, the mass-spring system will begin to oscillate.
	The displacement $y$ of the mass from its resting position is given by a function of the form
	\begin{center}
		$y(t)=c_1\cos(\omega t)+c_2\sin(\omega t)$
	\end{center}
	where $\omega$ is a constant that depends on the mass and the spring. (See the figure below.) Show that the set of all functions described above
	(with fixed $\omega$ and $c_1,c_2$ arbitrary) is a vector space.
	\begin{center}
		\includegraphics[scale=0.5]{Tema5/lay_4_1_19.eps}
	\end{center}
}{
  % Solution
	Let us call $V$ the set of all functions that can be expressed as
	\begin{center}
		$V=\{y(t)|y(t)=c_1\cos(\omega t)+c_2\sin(\omega t) \}$
	\end{center}
	To show that $V$ is a vector space we need to show that 
	$\forall \mathbf{u},\mathbf{v},\mathbf{w}\in V$ and $\forall c,d \in \mathbb{R}$
	\begin{enumerate}
		\item $\mathbf{u}+\mathbf{v}\in V$
		\item $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$
		\item $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$
		\item $\exists \mathbf{0}\in V | \mathbf{u}+\mathbf{0}=\mathbf{u}$
		\item $\forall\mathbf{u}\in V \quad \exists ! \mathbf{w}\in V | \mathbf{u}+\mathbf{w}=\mathbf{0}$ (we normally write $\mathbf{w}=-\mathbf{u}$)
		\item $c\mathbf{v}\in V$
		\item $c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}$
		\item $(c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}$
		\item $c(d\mathbf{u})=(cd)\mathbf{u}$
		\item $1\mathbf{u}=\mathbf{u}$
	\end{enumerate}
	Let's prove all these properties:
	\begin{enumerate}
		\item $\mathbf{u}+\mathbf{v}\in V$ \\
		      \begin{center}
						$\begin{array}{rcl}
							\mathbf{u}+\mathbf{v}&=&(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))+(c_{1v}\cos(\omega t)+c_{2v}\sin(\omega t))\\
							   &=&(c_{1u}+c_{1v})\cos(\omega t)+(c_{2u}+c_{2v})\sin(\omega t) \in V \\
						\end{array}$
					\end{center}
		\item $\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ \\
		      \begin{center}
						$\begin{array}{rcl}
							\mathbf{v}+\mathbf{u}&=&(c_{1v}\cos(\omega t)+c_{2v}\sin(\omega t))+(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
							   &=&(c_{1v}+c_{1u})\cos(\omega t)+(c_{2v}+c_{2u})\sin(\omega t)\\
							   &=&(c_{1u}+c_{1v})\cos(\omega t)+(c_{2u}+c_{2v})\sin(\omega t)=\mathbf{u}+\mathbf{v}\\
						\end{array}$
					\end{center}
		\item $(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$\\
		      \begin{center}
						$\begin{array}{rcl}
							(\mathbf{u}+\mathbf{v})+\mathbf{w}&=&\left((c_{1v}\cos(\omega t)+c_{2v}\sin(\omega t))+(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\right)+\\
							  & & (c_{1w}\cos(\omega t)+c_{2w}\sin(\omega t))\\
							&=&(c_{1u}+c_{1v})\cos(\omega t)+(c_{2u}+c_{2v})\sin(\omega t)+(c_{1w}\cos(\omega t)+c_{2w}\sin(\omega t))\\
							&=&(c_{1u}+c_{1v}+c_{1w})\cos(\omega t)+(c_{2u}+c_{2v}+c_{2w})\sin(\omega t)\\
						\end{array}$\\
						$\begin{array}{rcl}
							\mathbf{u}+(\mathbf{v}+\mathbf{w})&=&(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))+\\
							  & & \left((c_{1v}\cos(\omega t)+c_{2v}\sin(\omega t))\right)+(c_{1w}\cos(\omega t)+c_{2w}\sin(\omega t))\\
								&=& (c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))+(c_{1v}+c_{1w})\cos(\omega t)+(c_{2v}+c_{2w})\sin(\omega t)\\
								&=& (c_{1u}+c_{1v}+c_{1w})\cos(\omega t)+(c_{2u}+c_{2v}+c_{2w})\sin(\omega t)\\
								&=& (\mathbf{u}+\mathbf{v})+\mathbf{w}
						\end{array}$
					\end{center}
		\item $\exists \mathbf{0}\in V | \mathbf{u}+\mathbf{0}=\mathbf{u}$ \\
		      The addition neutral element is $\mathbf{0}=0\cos(\omega t)+0\sin(\omega t)$. Let's see why \\
		      \begin{center}
						$\begin{array}{rcl}
							\mathbf{u}+\mathbf{0}&=&(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))+(0\cos(\omega t)+0\sin(\omega t))\\
							   &=&(c_{1u}+0)\cos(\omega t)+(c_{2u}+0)\sin(\omega t)\\
							   &=&c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t)=\mathbf{u}\\
						\end{array}$
					\end{center}
		\item $\forall\mathbf{u}\in V \quad \exists ! \mathbf{w}\in V | \mathbf{u}+\mathbf{w}=\mathbf{0}$ (we normally write $\mathbf{w}=-\mathbf{u}$) \\
		      If $\mathbf{u}=c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t)$, its inverse with respect to addition is $-\mathbf{u}=-c_{1u}\cos(\omega t)-c_{2u}\sin(\omega t)$.
		      \begin{center}
						$\begin{array}{rcl}
							\mathbf{u}+(-\mathbf{u})&=&(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))+(-c_{1u}\cos(\omega t)-c_{2u}\sin(\omega t))\\
							   &=&(c_{1u}+(-c_{1u}))\cos(\omega t)+(c_{2u}+(-c_{2u}))\sin(\omega t)\\
							   &=&0\cos(\omega t)+0\sin(\omega t)=\mathbf{0}\\
						\end{array}$
					\end{center}
		\item $c\mathbf{v}\in V$ \\
		      \begin{center}
						$\begin{array}{rcl}
							c\mathbf{u}&=&c(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
							   &=&(cc_{1u})\cos(\omega t)+(cc_{2u})\sin(\omega t)\in V\\
						\end{array}$
					\end{center}
		\item $c(\mathbf{u}+\mathbf{v})=c\mathbf{u}+c\mathbf{v}$ \\
		      \begin{center}
						$\begin{array}{rcl}
							c(\mathbf{u}+\mathbf{v})&=&c((c_{1u}+c_{1v})\cos(\omega t)+(c_{2u}+c_{2v})\sin(\omega t))\\
							   &=&(cc_{1u}+cc_{1v})\cos(\omega t)+(cc_{2u}+cc_{2v})\sin(\omega t)\\
						\end{array}$\\
						$\begin{array}{rcl}
							c\mathbf{u}+c\mathbf{v}&=&(cc_{1u}\cos(\omega t)+cc_{2u}\sin(\omega t))+(cc_{1v}\cos(\omega t)+cc_{2v}\sin(\omega t))\\
							   &=& (cc_{1u}+cc_{1v})\cos(\omega t)+(cc_{2u}+cc_{2v})\sin(\omega t)=c(\mathbf{u}+\mathbf{v})\\
						\end{array}$\\
					\end{center}
		\item $(c+d)\mathbf{u}=c\mathbf{u}+d\mathbf{u}$ \\
		      \begin{center}
						$\begin{array}{rcl}
							(c+d)\mathbf{u}&=&(c+d)(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
							   &=&(c+d)c_{1u}\cos(\omega t)+(c+d)c_{2u}\sin(\omega t)\\
								 &=&(cc_{1u}+dc_{1u})\cos(\omega t)+(cc_{2u}+dc_{2u})\sin(\omega t)\\
								 &=&cc_{1u}\cos(\omega t)+dc_{1u}\cos(\omega t)+cc_{2u}\sin(\omega t)+dc_{2u}\sin(\omega t)\\
								 &=&(cc_{1u}\cos(\omega t)+cc_{2u}\sin(\omega t))+(dc_{1u}\cos(\omega t)+dc_{2u}\sin(\omega t))\\
								 &=&c(c_{1u}\cos(\omega t)+cc_{2u}\sin(\omega t))+d(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
								 &=&c\mathbf{u}+d\mathbf{u}\\
						\end{array}$\\
					\end{center}
		\item $c(d\mathbf{u})=(cd)\mathbf{u}$ \\
		      \begin{center}
						$\begin{array}{rcl}
							c(d\mathbf{u})&=&c(dc_{1u}\cos(\omega t)+dc_{2u}\sin(\omega t))\\
							   &=&cdc_{1u}\cos(\omega t)+cdc_{2u}\sin(\omega t)\\
							   &=&cd(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
							   &=&(cd)\mathbf{u}\\
						\end{array}$\\
					\end{center}
		\item $1\mathbf{u}=\mathbf{u}$ \\
		      \begin{center}
						$\begin{array}{rcl}
							1\mathbf{u}&=&1(c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t))\\
							   &=&(1\cdot c_{1u})\cos(\omega t)+(1\cdot c_{2u})\sin(\omega t)\\
							   &=&c_{1u}\cos(\omega t)+c_{2u}\sin(\omega t)\\
							   &=&\mathbf{u}\\
						\end{array}$\\
					\end{center}
	\end{enumerate}
}
\useproblem{lay:4_1_19}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
